Sensors & Transducers 2016 by IFSA Publishing, S. L.

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1 Sensors & ransducers 06 by IFSA Publishing, S. L. Polynomial Regression echniques or Environmental Data Recovery in Wireless Sensor Networks Kohei Ohba, Yoshihiro Yoneda, Koji Kurihara, akashi Suganuma, Hiroyuki Ito, Noboru Ishihara, Kunihiko Gotoh, Koichiro Yamashita, Kazuya Masu okyo Institute o echnology, Nagatsutacho 459, Midori-ku, Kanagawa, , Japan Network Systems Laboratory, Fujitsu Laboratories Ltd., Kamikodanaka 4, Kawasaki Nakahara-ku, Kanagawa, 8588, Japan el.: , ax: paper@lsi.pi.titech.ac.jp Received: 8 February 06 /Accepted: 5 April 06 /Published: 30 April 06 Abstract: In the near eature, large-scale wireless sensor networks will play an important role in our lives by monitoring our environment with large numbers o sensors. However, data loss owing to data collision between the sensor nodes and electromagnetic noise need to be addressed. As the interval o aggregate data is not ixed, digital signal processing is not possible and noise degrades the data accuracy. o overcome these problems, we have researched an environmental data recovery technique using polynomial regression based on the correlations among environmental data. he reliability o the recovered data is discussed in the time, space and requency domains. he relation between the accuracy o the recovered characteristics and the polynomial regression order is clariied. he eects o noise, data loss and number o sensor nodes are quantiied. Clearly, polynomial regression oers the advantage o low-pass iltering and enhances the signal-to-noise ratio o the environmental data. Furthermore, the polynomial regression can recover arbitrary environmental characteristics. Measured temperature and accelerator characteristics were recovered successully. Copyright 06 IFSA Publishing, S. L. Keywords: Wireless sensor networks, Data loss, Polynomial regression, Data recovery, Environment monitoring.. Introduction Large-scale wireless sensor networks (WSNs) use wireless sensor nodes to monitor environmental parameters such as temperature, humidity, ph, light and air pressure. WSNs have many possible applications, ranging rom structural health monitoring to ield monitoring. hanks to the progress in microelectronics based on the integrated circuit technology, small wireless sensor nodes with low power consumption have been achieved. However, problems exist with data loss owing to data collision between the sensor nodes and electromagnetic noise. As the interval o aggregate data is not ixed in the time and space domains, digital signal processing using Fourier or wavelet transorms cannot be applied directly to the aggregated data. Moreover, noise degrades the data accuracy. Because the environmental characteristics have various waveorms, data reliability cannot evaluated by signal

2 analysis. o overcome these problems, various techniques, such as data collection timing [], redundant system [], data recovery [3], have been used to increase data reliability. We apply polynomial regression to environmental data recovery based on the correlations among the environmental data [4]. Environmental characteristics are recovered as aggregated data rom the sensor nodes using polynomial regression. hus, data loss is tolerated, and the data can be analyzed easily. Basic sinusoidal environmental variations are assumed to evaluate the data recovery with polynomial regression. I the sinusoidal characteristics can be modeled appropriately, arbitrary waveorm characteristics, such as single-shot, periodic and non-periodic waveorms, can also be modelled theoretically. he recovered data accuracy is evaluated by comparing the recovered and source characteristics. We have also proposed a data reliability evaluation low that does not rely on signal analysis [5]. We also clariy the relation between the accuracy o the recovered characteristics and the polynomial regression order, and the eects o data loss and number o sensor nodes is analyzed. Furthermore, we show that the use o polynomial regression has the advantage o low-pass iltering that enhances the signal-to-noise ratio (SNR) o the environmental characteristics. In addition, we show that polynomial regression can recover arbitrary environmental characteristics. In Section, we introduce the environmental data recovery technique based on polynomial regression. In Section 3, the reliability o the recovered data is discussed. he requency domain characteristics are evaluated in Section 4. In Section 5, we conirm the ability o polynomial regression to recover arbitrary environmental characteristics. Results o data recovery or measured environmental characteristics are described in Section 5. Finally, the conclusions are drawn in Section 7.. Environmental Data Recovery Using Polynomial Regression he aggregated data analysis is shown in Fig.. I the interval o the aggregated sampled data is ixed, the environmental data characteristics can be analyzed directly using Fourier or wavelet transorms. However, when the interval o the data is not ixed, the data cannot be directly analyzed. hereore, continuous environmental characteristics are recovered rom the aggregated data, and then, the ixed interval data are resampled rom the recovered characteristics. Fig.. WSNs system using polynomial regression [4]... Polynomial Regression here are several ways o expressing the recovered characteristics, e.g., Fourier series expansion, polynomial regression, interpolation and so on. Polynomial regression is simple and suitable or expressing continuous characteristics as it tolerates data loss. However, polynomial regression is not good at expressing characteristics with many inlection points. I the requency band is limited, the environmental characteristics at the limited bandwidth can be expressed using polynomial expressions. hereore, polynomial regression is used in environmental data recovery. When one-dimensional data are t = [ t ] and the environmental data are t N d = [ d d N ], the environmental source characteristics unction W (t) can be recovered and recovered unction R (t ) is given by R m i ( t) = a t, () i= 0 i

3 where a = [ a 0 a m ] is the coeicient vector. he value o a is obtained using at least-suquares methods. m is the order o polynomial equation. For twodimensional data obtained by sensor nodes arranged in coordinates (x,y ),., (x N,y N ) and coordinates x = [ x x N ] and y = [ y y N ], the recovered unction R ( x, y ) is R j k ( x, = a x y, jk j, k 0 j+ k m () where the coeicient vector a is the column vector. ( m + )( m + ) It s size is... Data Reliability Evaluation Flow... Evaluation Flow he reliability evaluation low is shown in Fig.. wo-dimensional data are assumed in the evaluation. he environmental source characteristic unction is W( xi ). o consider the eect o noise and data loss, we deine the sensor node model. When the noise is expressed as x, y ), the sampled data with noise N ( i i x, y ) can be expressed as S( i i x, y ) = ( x, y ) + ( x, y ) (3) S ( i i W i i N i i o evaluate the eect o data loss, the ollowing unction is added. O ( xi ) = S ( xi ) ( without data loss), (4) 0 ( with data loss) where O( xi ) represents the sampled data considering the eect o noise and data loss. he amount o data in O( xi ) decreases compared with the number o S( xi ). he error o the recovered data is deined as E ( x, = R( x, + W( x, (5) he data accuracy that is sampled at ixed intervals using the above continuous unctions is compared with the accuracy o the evaluated data. he root mean-square error (RMSE) at each comparison point is deined as data reliability. RMSE is given by E σ W RMSE = mean( ( x, ) 00(%) (6) In the evaluation, we carry out 000 iterations to minimize the eect o noise variation and data loss. Fig.. Data reliability evaluation low [5].... Parameter Setting o evaluate the data recovery reliability, the ollowing conditions are considered. ) Sinusoidal Environmental Characteristics he correlations among the actual environmental data are complex. However, to determine a generalized index, it is preerable to use simple data characteristics. In this study, a sinusoidal wave is assumed as the basic environmental characteristic because any arbitrary characteristic can be expressed as a linear combination o a sinusoidal wave. he ollowing equations are the sinusoidal unctions used or one- and two-dimensional data. W W A ( x, = APP t ( t) = sin( π + θ ), (7) L PP sin(π ( x x0 ) + ( y y 0 ) L + θ ), (8) where (x 0,y 0 ) is showing the position o the wave generation source, A pp is the peak-to-peak amplitude, L is the wavelength and θ is the phase. 3

4 ) Observation Region he observation region is the region where the polynomial regression is applied. he observation region is partitioned and then polynomial regression is applied to each partition. Each data recovery unction is joined to express the characteristics o the observation region. he partitions o the region are determined by the cycle (wavelength) o the highest requency o the environmental characteristics. 3) Number o Sensor Nodes he sensor nodes are arranged at equal intervals in the observation region, including the upper boundary. In the case o two-dimensional structures, the sensor nodes are set on a grid. he density o the sensor nodes is represented by the number o sensor nodes N in the observation region. When the analysis is in the time domain, the one-dimensional coordinate axis is evaluated with respect to the time axis. In this case, the number o sensor nodes in the observation region represents the number o sampled data. 4) Noise Electromagnetic noise generated at the sensor interace consisting o an ampliier and an analogueto-digital converter and electromagnetic noise in the environment mainly contribute to data noise. he SNR is deined by ollowing equation. var( S ) SNR= 0log0 (9) var( ) White noise (Gaussian noise) is added in the reliability evaluation. 5) Data Loss Data loss occurs because o data collisions or intermittent ailures in the wireless communication. o simulate the eect o data loss, we use a pseudorandom data generation technique in the evaluation. 3. Data Reliability o Periodic Characteristics he reliability o recovered data that are resampled rom the recovered unction is evaluated by comparing with the environmental source characteristics unction. he data reliability is evaluated using the conditions described in the previous secti on. Fig. 3 shows the sinusoidal signal that is assumed as the environmental characteristics o one- and twodimensional conditions. he SNR at each sensor node is set to be 40 db. hereore, the reerence position o the RMSE is determined as.0 %. When the number o sensor nodes is increased, the reerence position o the RMSE is 0.5 %. Without sensor node noise, the reerence position o the RMSE is 0.5 %. N (b) wo-dimensional case Fig. 3. Examples o the recovery unction. 3.. Application Range o the Polynomial Regression Firstly, the relation between the partition region cycle and RMSE was analyzed by changing the order o the polynomial without the sensor node noise. he number o sensor nodes is ten or the one-cycle partition region in the one-dimensional case and 0 0 or the one-cycle partition region in the twodimensional case. We also examined the ive-cycle partition region and monitored the maximum error. Results or the one- and two-dimensional sinusoidal signals (Fig. 3) are shown in Fig. 4. For 0.5 % error and one-cycle partition region, the order o the polynomial should be higher than seven or one- and two-dimensional signals. 3.. Eect o Sensor Node Number he number o sensor nodes is thought to strongly aect the data reliability. he relation between the number o sensor nodes and RMSE was analyzed when the case o SNR is 50, 40 and 30 db at each sensor node. And ninth-order polynomial was used in the analysis. he results or the one- and twodimensional cases are shown in Fig. 5. Obviously, the errors are reduced with the number o sensor nodes. he increasing number o sensor nodes reduced the RMSE owing to noise. Fig. 6 shows the results or the 4

5 required SNR at each sensor node when the RMSE is 0.5 %, 0.5 % and %. he precision o eachsensor node is improved by increasing the number o sensor nodes. (b) wo-dimensional case Fig. 4. Precision o polynomial regression. (b) wo-dimensional case Fig. 5. Required number o sensor nodes [5]. (b) wo-dimensional case Fig. 6. Sensor accuracy [5]. here are two ways to improve the data reliability. he irst is to increase the number o sensor nodes and the second is that sensor nodes should be high SNR. I the number o sensor nodes is increased our times, the RMSE decreases by 50 %. I the SNR o each sensor node is improved by 6 db, the RMSE decreases by 50 % Eect o Data Loss he relation between data loss rate and the RMSE was analyzed. A ninth-order polynomial and 40-dB SNR at each sensor node was assumed. he number o sensor nodes was selected to satisy the RMSE o 0.5 % and 0.5 %. 5

6 In the one-dimensional case, 36 and 49 nodes were selected or the analysis. In the two-dimensional cases, 5 and 84 nodes were selected in the evaluation. he results or the one- and two-dimensional cases are shown in Fig. 7. he RMSE increases with data loss rate, o course. However, by increasing the number o sensor nodes, the RMSE decreases. he number o sensor nodes satisies the RMSE o 0.5 % adequately, whereas the data loss rate is 60 % or RMSE o 0.5 % in the one-dimensional case and 65 % in the two-dimensional case. hese results suggest that a redundant system can enhance the data reliability by increasing the number o sensor nodes. data. he signal-to-noise and distortion ratio (SNDR) and spurious-ree dynamic range (SFDR) were evaluated. he SFDR is used to discuss the eect o harmonic distortion. When the undamental requency is 0, the number o FF points is FF and the sampling requency POIN is F S, the 0 is given by FS 0 = (0) FF POIN hereore, the input signal requency in and wavelength o the input signal per division λ is in in = m 0, () m λ in =, () D n where D is the number o divisions and m is an n integer number. 4.. Reliability in the FF Analysis 4... Eect o Input Signal Cycle (Wavelength) he relation between signal cycle (wavelength) in the polynomial regression and the evaluation indices o gain, SNDR and SFDR was analyzed using FF. In the analysis, a ninth-order polynomial, 40-dB SNR at each sensor node, 3 divisions dividing FF points into partition region and 04 o FF points are assumed. Evaluattion conditions are summarized in able. (b) two-dimensional case Fig. 7. Data loss robustness [5]. 4. Reliability in the Frequency Domain he reliability o the recovered data using polynomial regression was also evaluated in the requency domain [6]. he ast Fourier transorm (FF) was applied to the recovered data. 4.. FF Analysis he one-dimensional sinusoidal environmental characteristics are assumed to be the same as in the previous sections. Recovered data at ixed intervals are obtained by sampling the data recovered by polynomial regression. FF is applied to the recovered able. Conditions o FF analysis [6]. Parameters Conditions Noise 40 [db] Polynomial regression order 9 FF points 04 Divisions 3 he results are shown in Fig. 8. his is showing the relation between input requency cycle (wavelength) or polynomial regression and the evaluation indexes. Decreases o db are tolerated by the SNDR and SFDR and or wavelength with the maximum partition o.6 cycles. Above.6 cycles, the partition region signals are iltered out. he gain is lat up to the threecycle partition region. hus, the polynomial regression acts as a low-pass ilter. his means that the SNDR and SFDR improve because the polynomial regression limits the bandwidth o the environmental signals. 6

7 Fig. 8. SNDR, SFDR and gain vs. input wavelength [5] Eect o the Number o Sensor Node he number o sensor nodes per partition region is evaluated. he results are shown in Fig. 9. he FF results or the source environmental signals were 40- db SNDR and 59-dB SFDR. For 3 sensor nodes, the SNDR is the same as the result o the source environmental signals. For 5 sensor nodes, the SNDR is the same as the result o source environmental signals. Higher SNDR and SFDR are possible by increasing the number o sensor nodes. By increasing the number o sensor nodes our times, both SNDR and SFDR improved by 6 db. Fig. 9. SNDR and SFDR vs. number o sensor nodes [6] Frequency Spectrum he requency spectrum is evaluated by FF. he results are shown in Fig. 0. Based on the results o Figs. 8 and 9, the.6-cycle (wavelength) input signal region and 5 sensor nodes per partition region were assumed. Compared with the spectrum o the source environmental signal, the noise level o the highrequency region is iltered out. able summarizes conditions o the FF analysis and able summarizes the result o FF analysis. By limiting the observation region in the polynomial regression, both SNDR and SFDR are improved. Fig. 0. Frequency spectrum with and without recovered data [6]. able. Result o FF analysis [6]. SNDR [db] SFDR [dbc] (a) FF (b) FF (using recovered data) Dierences ((b)-(a)) Arbitrary Characteristics Recovery by Polynomial Regression In Sections 3 and 4, it was clariied that polynomial regression can recover the sinusoidal environmental characteristics. Polynomial regression or arbitrary characteristics is also validated by selecting the order o the polynomial equation or each partition region. Scale-space iltering (SSF) [7] and Akaike s inormation criterion (AIC) [8] were used to select the partition region and the order o the polynomial regression based on aggregate data. he SSF detects extreme values by the convolution o the Gaussian unction. he partition region is obtained as the region between the extreme points. he AIC is a statistical measure that estimates the quality o the environmental source characteristics rom aggregate data, including noise and data loss. he order o the polynomial regression or the partition region is obtained by the SSF. By detecting the extreme values by SSF and estimating the quality o source characteristics using polynomial regression between the appropriately selected extreme points by AIC, arbitrary characteristics can be recovered [9]. Fig. shows extreme values o arbitrary environmental characteristics with 40-dB SNR detected by SSF. he observation region is divided into partition regions using the extreme values and the order o polynomial regression is thus optimized. he regions divided by the criterion o extreme values are the partition regions, and the order o the polynomial regression is set at each partition region. Fig. shows the recovered data rom arbitrary characteristics with 40-dB SNR using the SSF, AIC and polynomial regression. he RMSE is under 0.%; thus, the polynomial regression can obviously recover the arbitrary environmental characteristics. 7

8 his result shows the data recovery with the polynomial regression is applicable or the environmental data that changes successively at random. Fig.. Extreme values o arbitrary characteristics [4]. Fig. 3. Measured temperature characteristics. able 3. Data recovery conditions or measured temperature. Fig.. Data recovery unction based on arbitrary characteristics [4]. Parameters Conditions erm rom Jul. 7 th, 05 to Aug. 6 th, 05 Interval o Sampling 0 minutes Number o the partition region 60 Number o samples (/one partition region) 36 Polynomial order 8 6. Recovery or Measured Data by Polynomial Regression In Sections 5, it was clariied that polynomial regression can recover the arbitrary environmental characteristics. hereore, polynomial regression was applied to the actual measured data. wo examples are shown in the section. 6.. emperature Data in ime Domain Example o measured temperature data is shown in Fig. 3. he data was sampled at every 0 minutes rom Jul. 7 th, 05 to Aug. 6 th, 05 ( month) using thermos recorder placed in a arm. he temperature change which is day by day is observed. Polynomial regression techniques previously described were applied to the measured data o every hours. he polynomial order is the same or the each partition regions, and the polynomial order was optimized to be the minimum value o the RMSE in each the polynomial order. able 3 summarizes the conditions or the temperature data recovery. he recovered results are shown in Fig. 4. Measured characteristics in Fig. 3 were recovered with the RMSE less than about ± %. Fig. 4. Recovered temperature characteristics. 6.. Acceleration Data in Frequency Domain Recovery o measured acceleration sensor data was also examined with the polynomial regression techniques. Recovered requency spectrum was compared with measured characteristics. Acceleration sensor data was measured with 0-Hz signal vibrator supposing person s health monitoring. he sampling requency o acceleration sensor was 50 Hz. Polynomial regression was applied to every 3 data o the acceleration sensor data measured. 8

9 he results and data recovery conditions or measured acceleration are shown in Fig. 5 and able 4 respectively. he LPF eect is observed in the data recovered. And in the low requency region including the 0-Hz signal, it can be seen that there is no dierence between the power spectrums measured and power spectrums recovered. polynomial regression enhances the SNDR or SFDR in the WSNs system. 4) Polynomial regression can recover arbitrary environmental characteristics and can be used with SSF and AIC. 5) Proposed polynomial regression techniques were successully applied to two kinds o measured data; temperature and acceleration. In conclusion, environmental data recovery using polynomial regression can be applied to wireless sensor networks. Acknowledgements Part o this work was a joint research project o okyo Institute o echnology and Fujitsu Laboratories Limited. We would like to express our thanks to all who supported this project. Reerences Fig. 5. Data recovery or measured acceleration. he eect o the polynomial regression in the requency domain was conirmed with the acceleration sensor data measured. able 4. Data recovery conditions or measured acceleration. Parameters Conditions Input requency 0 [Hz] Sampling requency 50 [Hz] Number o the partition region 3 Number o samples (/one partition region) 3 Polynomial order 3 7. Conclusions In this paper, environment data recovery techniques using polynomial regression or wireless sensor networks (WSNs) have been examined and discussed. ) A data reliability evaluation procedure or WSNs was proposed with polynomial regression. ) he recovered data reliability depends on the order o the polynomial regression; the number o sensor nodes, the eect o noise and data loss were quantiied. 3) From FF analysis, it is seen that polynomial regression act as a low-pass ilter. Data recovery using []. Sivrikaya F., et al., ime synchronization in sensor networks: a survey, IEEE Network, Vol. 8, No. 4, 004, pp []. K. Yamashita, et al., Implementation and Evaluation o Architecture Search Simulator Including Disturbance or Wide-range Grid Wireless Sensor Network, in Proceedings o the Multimedia, Distributed, Cooperative and Mobile Symposium, 04, pp [3]. Doherty L., et al., Algorithms or position and data recovery in wireless sensor networks, Diss. Department o Electrical Engineering and Computer Sciences, University o Caliornia at Berkeley, 000. [4]. K. Ohba, et al., Environmental Data Recovery Using Polynomial Regression or Large-scale Wireless Sensor Networks, in Proceedings o the 5 th International Conerence on Sensor Networks (SENSORNES 6), No., 06, pp [5]. Y. Yoneda, et al., A study on Data Reliability Evaluation Index o Wireless Sensor Network or Environmental Monitoring, in Proceedings o the 4 st SICE Symposium on Intelligent Systems, 04. [6]. K. Ohba, et al., A method o recovering environment data using polynomial regression or large-scale wireless sensor networks, IEICE echnical Report, ASN05-7, 05, pp. -6. [7]. A. P. Witkin, Scale-Space Filtering: A New Approach o Multi-Scale Description, in Proceedings o the IEEE International Conerence on Acoustics, Speechand Signal Processing, San Diego, CA, Vol. 9, 984, pp [8]. H. Akaike, A New Look at the Statistical Model Identiication, IEEE ransactions on Automatic Control, Vol. AC-9, No. 6, 974, pp [9]. K. Haze, et al., Modeling home appliance power consumption by interval-based switching Kalman ilters, IEICE echnical Report, Vol., No. 3, 0, pp Copyright, International Frequency Sensor Association (IFSA) Publishing, S. L. All rights reserved. ( 9